用于符号数学的 Python 库——sympy(二)：常用信号的Laplace变换

关于拉普拉斯变换的作用，可参考知乎
总的来说，拉普拉斯变换就是迫使函数满足绝对可积条件的傅里叶变换。

常用信号的Laplace变换

参考信号与系统/陈后金, 胡健, 薛健. ——2版.——北京：清华大学出版社；北京交通大学出版社, 2005.7(2017.3重印)第218-219页。

序号	单边信号(f(t))	Laplace变换( $F (s)$ )	收敛域
1	$e^{-\lambda t}u(t)$	$\frac{1}{s+\lambda}$	$Re(s)>-\lambda$
2	$e^{j\omega_0 t}u(t)$	$\frac{1}{s-j\omega_0}$	$R e (s) > 0$
3	$cos(\omega_0 t)u(t)$	$\frac{s}{s^2+\omega_0^2}$	$R e (s) > 0$
4	$sin(\omega_0 T)u(t)$	$\frac{\omega_0}{s^2+\omega_0^2}$	$R e (s) > 0$
5	$e^{-\sigma_0 t}cos(\omega t)u(t)$	$\frac{s+\sigma_0}{(s+\sigma_0)^2+\omega_0}$	$Re(s)>-\sigma_0$
6	$e^{-\sigma_0 t}sin(\omega t)u(t)$	$\frac{\omega_0}{(s+\sigma_0)^2+\omega_0}$	$Re(s)>-\sigma_0$
7	$\delta(t)$	1	$Re(s)>-\infty$
8	$\delta^{(n)}(t)$	$s^n(1,2,\dots)$	$Re(s)>-\infty$
9	$u (t)$	$\frac{1}{s}$	$R e (s) > 0$
10	$t u (t)$	$\frac{1}{s^2}$	$R e (s) > 0$
11	$t^n u(t)$	$\frac{n!}{s^{n+1}}$	$R e (s) > 0$
12	$te^{-\lambda t}u(t)$	$\frac{1}{(s+\lambda)^2}$	$Re(s)>-\lambda$

1. $e^{-\lambda t}u(t)$

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

lamb = symbols('\lambda')
F1 = laplace_transform(exp(-lamb*t)*Heaviside(t),t,s)

F1[0]
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$\displaystyle \frac{1}{\lambda + s}$

F1[1]
1

$\displaystyle - \lambda$

2. $e^{j\omega_0 t}u(t)$

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

w0 = symbols('\omega_0')
F1 = laplace_transform(exp(np.complex(0,1)*w0*t)*Heaviside(t),t,s)

F1[0]
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$\displaystyle \frac{1}{- 1.0 i \omega_{0} + s}$

3. $cos(\omega_0 t)u(t)$

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

w0 = symbols('\omega_0')
F1 = laplace_transform(cos(w0*t)*Heaviside(t),t,s)

F1[0]
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2
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$\displaystyle \frac{s}{\omega_{0}^{2} + s^{2}}$

4. $sin(\omega_0 T)u(t)$

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

w0 = symbols('\omega_0')
F1 = laplace_transform(sin(w0*t)*Heaviside(t),t,s)

F1[0]
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$\displaystyle \frac{\omega_{0}}{\omega_{0}^{2} + s^{2}}$

5. $e^{-\sigma_0 t}cos(\omega t)u(t)$

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

w0,sigma0 = symbols('\omega_0 \sigma_0')
F1 = laplace_transform(exp(-sigma0*t)*cos(w0*t)*Heaviside(t),t,s)

F1[0]
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$\displaystyle \frac{\sigma_{0} + s}{\left(- i \omega_{0} + \sigma_{0} + s\right) \left(i \omega_{0} + \sigma_{0} + s\right)}$

6. $e^{-\sigma_0 t}sin(\omega t)u(t)$

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

w0,sigma0 = symbols('\omega_0 \sigma_0')
F1 = laplace_transform(exp(-sigma0*t)*sin(w0*t)*Heaviside(t),t,s)

F1[0]
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2
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7

$\displaystyle \frac{\omega_{0}}{\left(- i \omega_{0} + \sigma_{0} + s\right) \left(i \omega_{0} + \sigma_{0} + s\right)}$

7. $\delta(t)$

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

F1 = laplace_transform(DiracDelta(t),t,s)

F1[0]
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$\displaystyle 1$

8. $\delta^{(n)}(t)$

n次求导无法计算，用多个数字代替，总结规律(不严谨) 结果看不懂

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

delta,n = symbols('\delta n')
F1 = laplace_transform(DiracDelta(t,2),t,s)

F1
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$\displaystyle \mathcal{L}_{t}\left[\delta^{\left( 2 \right)}\left( t \right)\right]\left(s\right)$

9. $u (t)$

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

delta,n = symbols('\delta n')
F1 = laplace_transform(Heaviside(t),t,s)

F1[0]
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$\displaystyle \frac{1}{s}$

10. $t u (t)$

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

delta,n = symbols('\delta n')
F1 = laplace_transform(t*Heaviside(t),t,s)

F1[0]
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2
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7

$\displaystyle \frac{1}{s^{2}}$

11. $t^n u(t)$

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

delta,n = symbols('\delta n')
F1 = laplace_transform(t**n*Heaviside(t),t,s)

F1[0]
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$\displaystyle \frac{s^{- n} \Gamma\left(n + 1\right)}{s}$

12. $te^{-\lambda t}u(t)$

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

lamb,n = symbols('\lambda n')
F1 = laplace_transform(t*exp(-lamb*t)*Heaviside(t),t,s)

F1[0]
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2
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$\displaystyle \frac{1}{\left(\lambda + s\right)^{2}}$

至于 $\Gamma$ 函数
$\Gamma(x)=\int_0^{+\infty}t^{x-1}e^{-t}dt$
当 $x$ 为正整数时，
$\Gamma(n)=(n-1)!$

逆Laplace变换

逆Laplace变换与Laplace变换函数用法大致一致，举一个简单例子，对于 $e^{-\lambda t}u(t)$ 的拉普拉斯变换，输出结果为
$\frac{1}{\lambda+s}$ ,那么对于 $\frac{1}{\lambda+s}$ 的逆拉普拉斯变换，代码为

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

lamb,s,t = symbols('\lambda s t')
F1 = inverse_laplace_transform(1/(lamb+s),s,t)

F1
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$\displaystyle e^{- t \operatorname{re}{\left(\lambda\right)} - i t \operatorname{im}{\left(\lambda\right)}} \theta\left(t\right)$
结果的 $\theta(t)$ 就是 $u (t)$

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原文地址：https://blog.csdn.net/u011740601/article/details/127883982

用于符号数学的 Python 库——sympy(二)：常用信号的Laplace变换

常用信号的Laplace变换

1. e − λ t u ( t ) e^{-\lambda t}u(t) e−λtu(t)

2. e j ω 0 t u ( t ) e^{j\omega_0 t}u(t) ejω0​tu(t)

3. c o s ( ω 0 t ) u ( t ) cos(\omega_0 t)u(t) cos(ω0​t)u(t)

4. s i n ( ω 0 T ) u ( t ) sin(\omega_0 T)u(t) sin(ω0​T)u(t)

5. e − σ 0 t c o s ( ω t ) u ( t ) e^{-\sigma_0 t}cos(\omega t)u(t) e−σ0​tcos(ωt)u(t)

6. e − σ 0 t s i n ( ω t ) u ( t ) e^{-\sigma_0 t}sin(\omega t)u(t) e−σ0​tsin(ωt)u(t)

7. δ ( t ) \delta(t) δ(t)

8. δ ( n ) ( t ) \delta^{(n)}(t) δ(n)(t)

9. u ( t ) u(t) u(t)

10. t u ( t ) tu(t) tu(t)

11. t n u ( t ) t^n u(t) tnu(t)

12. t e − λ t u ( t ) te^{-\lambda t}u(t) te−λtu(t)